It says Chelsea - which fired Mourinho early into the 2015/16 season after a poor start - could demand a "six-figure sum" for the trademark before United officially signs him.
Chelsea reportedly registered the name "Jose Mourinho," as well as his signature, as a European trademark back in 2005.
Despite Mourinho's sacking, the club still owns and uses those trademarks on much of its official merchandise.
United announced the dismissal of manager Louis van Gaal on Monday and has been in talks with Mourinho since.
It was expected that Mourinho's appointment as the club's new manager would be confirmed on Tuesday, but there has yet to be any official announcement.
The Guardian says that Ed Woodward, vice chairman of Manchester United, wants to acquire Mourinho's image rights as they will provide a lucrative income. If he can't, he may have to license the image rights from Chelsea - a very strange state of affairs.
The issue will be resolved either way, it adds, and Mourinho will sign a three-year deal with the club in excess of his £7.5 million ($11 million) annual salary at Chelsea.
In related news, an employment tribunal against Mourinho and Chelsea by Eva Carneiro - a sports medicine specialist who was fired from Chelsea's first-team duties last season - will go ahead in June.
Women in Football - a campaign group - previously said it was "appalled" at the Football Association's decision to clear Mourinho for discriminating comments he allegedly made against Carneiro.